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Theorem sbel2x 2299
Description: Elimination of double substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 29-Sep-2018.)
Assertion
Ref Expression
sbel2x  |-  ( ph  <->  E. x E. y ( ( x  =  z  /\  y  =  w )  /\  [ y  /  w ] [
x  /  z ]
ph ) )
Distinct variable group:    x, y,
ph
Allowed substitution hints:    ph( z, w)

Proof of Theorem sbel2x
StepHypRef Expression
1 nfv 1772 . . 3  |-  F/ y
ph
2 nfv 1772 . . 3  |-  F/ x ph
31, 22sb5rf 2291 . 2  |-  ( ph  <->  E. y E. x ( ( y  =  w  /\  x  =  z )  /\  [ y  /  w ] [
x  /  z ]
ph ) )
4 ancom 456 . . . 4  |-  ( ( y  =  w  /\  x  =  z )  <->  ( x  =  z  /\  y  =  w )
)
54anbi1i 706 . . 3  |-  ( ( ( y  =  w  /\  x  =  z )  /\  [ y  /  w ] [
x  /  z ]
ph )  <->  ( (
x  =  z  /\  y  =  w )  /\  [ y  /  w ] [ x  /  z ] ph ) )
652exbii 1730 . 2  |-  ( E. y E. x ( ( y  =  w  /\  x  =  z )  /\  [ y  /  w ] [
x  /  z ]
ph )  <->  E. y E. x ( ( x  =  z  /\  y  =  w )  /\  [
y  /  w ] [ x  /  z ] ph ) )
7 excom 1938 . 2  |-  ( E. y E. x ( ( x  =  z  /\  y  =  w )  /\  [ y  /  w ] [
x  /  z ]
ph )  <->  E. x E. y ( ( x  =  z  /\  y  =  w )  /\  [
y  /  w ] [ x  /  z ] ph ) )
83, 6, 73bitri 279 1  |-  ( ph  <->  E. x E. y ( ( x  =  z  /\  y  =  w )  /\  [ y  /  w ] [
x  /  z ]
ph ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 375   E.wex 1674   [wsb 1808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1675  df-nf 1679  df-sb 1809
This theorem is referenced by: (None)
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